Problem: Let $a$ and $b$ be integers such that $ab = 100.$  Find the minimum value of $a + b.$
We claim that the minimum value is $-101.$

If $a = -1$ and $b = -100,$ then $ab = 100$ and $a + b = -101.$

Now,
\begin{align*}
a + b + 101 &= a + \frac{100}{a} + 101 \\
&= \frac{a^2 + 101a + 100}{a} \\
&= \frac{(a + 1)(a + 100)}{a}.
\end{align*}If $a$ is positive, then $b$ is positive, so $a + b$ is positive, so suppose $a$ is negative.  Then $b$ is negative.  Furthermore, since $a$ is a factor of 100, $-100 \le a \le -1.$  Hence, $a + 1 \le 0$ and $a + 100 \ge 0,$ so
\[a + b + 101 = \frac{(a + 1)(a + 100)}{a} \ge 0.\]Equality occurs if and only if $a = -1$ or $a = -100,$ both of which lead to $a + b = -101.$

Therefore, the minimum value of $a + b$ is $\boxed{-101}.$